Based on my research, but no deep implementation experience with either:

XMPP continues to advance, so could be considered modern if you use the latest versions.

Matrix is more of an event sync protocol, consisting of a server to server sync protocol and a client to server protocol (sort of like email). It is mostly used for chat, but appears to be more flexible. Matrix uses JSON and HTTP.

XMPP is focused on chatmessages. In many ways XMPP and Matrix are similar. XMPP is based on XML and originally used TCP as the transport protocol, but has been extended to use HTTP now as well.

EDIT: I think a clearer way to describe the basic difference is that Matrix syncs Matrix Events across servers, whereas XMPP facilitates exchanging messages with clients. Most folks use the protocols to exchange chat messages. It is the way this is accomplished by the corresponding servers that differs significantly.